Abstract

We extend the well-known analogy between the problems of paraxial diffraction in space and dispersion in time to optical pulse compression and propose a time-domain analog to spatial imaging that allows for the distortionless expansion or compression of optical power waveforms. We call this new concept temporal imaging and derive equivalent expressions for the focal length and the f-number of a time lens and the magnification of an imaging system. It should now become possible, with a temporal microscope, to expand ultrafast optical phenomena to a time scale that is accessible to conventional high-speed photodiodes.

© 1989 Optical Society of America

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References

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  1. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
    [CrossRef]
  2. D. Grischkowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
    [CrossRef]
  3. D. Grischkowsky, Appl. Phys. Lett. 25, 566 (1974).
    [CrossRef]
  4. B. H. Kolner, Appl. Phys. Lett. 52, 1122 (1988).
    [CrossRef]
  5. E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. M. Haner, W. S. Warren, in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, S. Shionoya, eds. (Springer-Verlag, Berlin, 1988).
  8. P. Maine, G. Mourou, Opt. Lett. 13, 467 (1988).
    [CrossRef] [PubMed]
  9. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  10. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).
  11. M. Nazarathy, J. Shamir, J. Opt. Soc. Am. 70, 150 (1980).
    [CrossRef]

1988 (2)

1982 (1)

D. Grischkowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

1980 (1)

1974 (1)

D. Grischkowsky, Appl. Phys. Lett. 25, 566 (1974).
[CrossRef]

1969 (1)

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

1968 (1)

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Balant, A. C.

D. Grischkowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Drabovich, K. N.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grischkowsky, D.

D. Grischkowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

D. Grischkowsky, Appl. Phys. Lett. 25, 566 (1974).
[CrossRef]

Haner, M.

M. Haner, W. S. Warren, in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, S. Shionoya, eds. (Springer-Verlag, Berlin, 1988).

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Khokhlov, R. V.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Kolner, B. H.

B. H. Kolner, Appl. Phys. Lett. 52, 1122 (1988).
[CrossRef]

Kovrigin, A. I.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Maine, P.

Mourou, G.

Nazarathy, M.

Shamir, J.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

Sukhorukov, A. P.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Treacy, E. B.

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

Warren, W. S.

M. Haner, W. S. Warren, in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, S. Shionoya, eds. (Springer-Verlag, Berlin, 1988).

Appl. Phys. Lett. (3)

D. Grischkowsky, A. C. Balant, Appl. Phys. Lett. 41, 1 (1982).
[CrossRef]

D. Grischkowsky, Appl. Phys. Lett. 25, 566 (1974).
[CrossRef]

B. H. Kolner, Appl. Phys. Lett. 52, 1122 (1988).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. B. Treacy, IEEE J. Quantum Electron. QE-5, 454 (1969).
[CrossRef]

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Other (4)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Haner, W. S. Warren, in Ultrafast Phenomena VI, T. Yajima, K. Yoshihara, C. B. Harris, S. Shionoya, eds. (Springer-Verlag, Berlin, 1988).

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Figures (2)

Fig. 1
Fig. 1

(a) Conventional fiber-grating pulse compressor and (b) its spatial analog, a lens Fourier transform.

Fig. 2
Fig. 2

Space-time imaging analogy. (a) Temporal imaging configuration. (b) Spatial imaging configuration.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

E ( z , t ) = u ( z , t ) exp j ( ω t β z )
u ξ = j 2 2 β ω 2 2 u τ 2 ,
u ( ξ , τ ) = 1 2 π + U ( 0 , ω ) exp ( j ξ 2 2 β ω 2 ω 2 ) exp ( j ω t ) d ω .
ϕ ( t ) = ± A ( 1 ω m 2 t 2 2 ) ,
f T = ω 0 A ω m 2 .
f # = f T ω m = ω 0 A ω m .
u ( τ , ξ ) = ( 1 M ) 1 / 2 exp ( j A ω m 2 τ 2 2 M ) + U ( 0 , ω ) × exp [ j ω 2 ( 1 ξ 1 2 β 1 ω 2 + 1 ξ 2 2 β 2 ω 2 ω 0 f T ) ] exp ( j ω τ M ) d ω ,
1 ξ 1 2 β 1 ω 2 + 1 ξ 2 2 β 2 ω 2 = ω 0 f T .

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